Breaking of stationary waves in nonlinear dispersive media

A nontrivial behaviour of a nonlinear wave under influence of small disturbing factors like dissipation, smooth inhomogeneity of medium parameters, etc. is considered by the example of sine-Gordon equation. The stage of slow “adiabatic” variation of the parameters of quasi-stationary wave is shown to change at some finite distance due to strong instability. The wave form becomes essentially nonstationary (breaking of stationary wave structure). The breaking condition is defined by the extremum of the wave adiabatic invariant. The behaviour of a wave at the nonadiabatic stage is described using a Galerkin procedure.     

Another possible model equation for long waves in nonlinear dispersive systems

In the same spirit in which Benjamin, Bona, and Mahoney modified the Korteweg-de Vries equation (U_x+U_t+UU_x+U_xxx=0) to obtain the so-called BBM equation, U_x+U_t+UU_x?U_xxt=0, we propose a different modification: U_x+U_t+UU_x+U_xtt=0. The advantages in this equation are 1) the system is conservative since it can be derived from the Lagrangian density $\text{L=}\text{1}\text{2}\text{θ}{}_{\mathrm{x}}\text{θ}{}_{\mathrm{t}}\text{+}\text{1}\text{2}\text{θ}{}^2{}_{\mathrm{x}}\text{+}\text{1}\text{6}\text{θ}{}^3{}_{\mathrm{x}}\text{?}\text{1}\text{2}\text{θ}{}^2{}_{\mathrm{xt}}\text{,}\text{where}\text{θ}{}_{\mathrm{x}}\text{≡ U;2)}$ for large wave-numbers |k|, the infinitesimal-wave phase speed falls off like $\text{1}\text{|k|}$, in accord with physical intuition; 3) since the equation is of second order in t, both U and U_t can be independently specified for t = 0. Several conservation laws satisfied by solutions to this equation are given.     

Secular-free solution up to third order of a model nonlinear equation for dispersive waves

The perturbation method of Lindstedt is applied to study the non linear effect of a nonlinear equation $$\nabla ^2 {\rm E} - \frac{1}{{c^2 }}\frac{{\partial ^2 {\rm E}}}{{\partial t^2 }} - \frac{{\omega _0^2 }}{{c^2 }}{\rm E} + \frac{{2v}}{{c^2 }}\frac{{\partial {\rm E}}}{{\partial t}} + E^2 \left[ {\frac{{\partial {\rm E}}}{{\partial t}} \times A} \right] = 0,$$ where (A. E)=0 andA,c, ω ₀ andν are constants in space and time. Amplitude dependent frequency shifts and the solution up to third order are derived.     

Nonlocal models for nonlinear, dispersive waves

Considered herein are model equations for the unidirectional propagation of small-amplitude, nonlinear, dispersive, long waves such as those governed by the classical Korteweg-de Vries equation. Of special interest are physical situations in which the linear dispersion relation is not appropriately approximated by a polynomial, so that the operator modelling dispersive effect is nonlocal. Particular cases in view here are the Benjamin-Ono equation and the intermediate long-wave equation which arise in internal-wave theory, and Smith's equation which governs certain types of continental-shelf waves.

The initial-value problem for these equations is shown to be globally well posed in the classical sense, including continuous dependence upon the initial data and, in certain cases upon the modelling of nonlinear and dispersive effects. Whilst the results are stated for the specific equations listed above, the techniques utilized are seen to have a considerable range of generality as regards application to nonlinear, dispersive evolution equations. Particularly worthy of note is our theorem implying that solutions of the intermediate long-wave equation converge strongly to solutions of the Korteweg-de Vries equation, or to solutions of the Benjamin-Ono equation, in appropriate asymptotic limits.     

Energy propagation for dispersive waves in dissipative media

University of Bologna, Italy. Published in Izvestiya Vysshikh Uchebnykh Zavedenii, Radiofizika, Vol. 36, No. 7, pp. 650–664, July, 1993.     

一类非线性色散方程中的新型奇异孤立波

研究一类非线性色散广义DGH方程的新型奇异孤立波及其Painlevé可积性.利用Painlevé分析发现当对流项强度m=2时广义DGH方程是可积的,这是一个新的可积方程.通过构造新的变量代换以及auto-Backlund变换获得该方程丰富的奇异孤立波解,如紧孤立波（compacton）、尖峰孤立波（peakon）、新型带尖点的双孤立波和带爆破点的双孤立波等. 关键词： 非线性色散方程 可积性 奇异孤立波     

Self-precession and frequency shifts for three interacting waves for a nonlinear model equation in a dispersive medium

The wave-wave interaction of three interacting waves for a nonlinear model equation in a dispersive medium is studied. It is found that the nonlinear effect causes precession of the polarization ellipse as the wave propagates under the resonance condition ₁ = ₂ + ₃ and k₁ k₂ + k₃. The frequency shifts and the precessional frequencies are obtained under the above resonance conditions. A technique has been developed for the evaluation of self-precession and frequency shifts for circularly polarized waves.The author expresses his gratitude to Dr. B. Chakraborty, Department of Mathematics, Jadavpur University, Calcultta for his guidance in the preparation of this paper and also to the referee for his suggestion in the improvement of this paper.     

Nonlinear waves in weakly dispersive random media

The propagation of nonlinear waves in random media is an important aspect of nonlinear wave theory and has a long and informative history. This paper describes the basic ideas of the approaches that have been applied. The average-field method, which has been used most extensively in linear problems, is considered. This approach is then shown to be incorrect as far as nonlinear processes are concerned. Finally, a new scheme is proposed average-form the method, which allows consistent evolution equations to be obtained for nonlinear waves in random media.Institute of Applied Physics, Russian Academy of Sciences. Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Radiofizika, Vol. 36, No. 8, pp. 760–766, August, 1993.     

Hydrodynamics of modulated finite-amplitude waves in dispersive media

Analytic approaches are developed for integrating the nondiagonalizable Whitham equations for the generation and propagation of nonlinear modulated finite-amplitude waves in dissipationless dispersive media. Natural matching conditions for these equations are stated in a general form analogous to the Gurevich-Pitaevskii conditions for the averaged Korteweg-de Vries equations. Exact relationships between the hydrodynamic quantities on different sides of a dissipationless shock wave, an analog of the shock adiabat in ordinary dissipative hydrodynamics and first proposed on the basis of physical considerations by Gurevich and Meshcherkin, are obtained. The boundaries of a self similar, dissipationless shock wave are determined analytically as a function of the density jump. Some specific examples are considered. Zh. éksp. Teor. Fiz. 115, 1116–1136 (March 1999)     

Nearly linear dynamics of nonlinear dispersive waves

Dispersive averaging effects are used to show that the Korteweg-de Vries (KdV) equation with periodic boundary conditions possesses high frequency solutions, which behave nearly linearly. Numerical simulations are presented, which indicate the high accuracy of this approximation. Furthermore, this result is applied to shallow water wave dynamics in the limit of KdV approximation, which is obtained by asymptotic analysis in combination with numerical simulations of KdV.     

分层介质半空间瑞利波的时频分析

对分层介质半空间瑞利波的频散特性,用一种时频分析方法——重排的平滑伪魏格纳维尔分布(RSPWVD,Reassignment of Smooth Pseudo-Wigner Ville Distribution)进行了分析和研究。对均匀半空间和两层介质半空间的理论和实验研究表明,由于层状介质中瑞利波的频散曲线存在多个模式,所获得的群速度频散曲线在不同的频段显示出来的模式是对应表面位移幅度占主导作用的模式。频散曲线的这种模式判定对利用层状半空间的瑞利波反演介质参数是必须预先了解的。     

Local diffusion theory for localized waves in open media

We report a first-principles study of static transport of localized waves in quasi-one-dimensional open media. We find that such transport, dominated by disorder-induced resonant transmissions, displays novel diffusive behavior. Our analytical predictions are entirely confirmed by numerical simulations. We show that the prevailing self-consistent localization theory [B. A. van Tiggelen, Phys. Rev. Lett. 84, 4333 (2000)] is valid only if disorder-induced resonant transmissions are negligible. Our findings open a new direction in the study of Anderson localization in open media.     

Spatio-temporal pulse propagation in nonlinear dispersive optical media

We discuss state-of-art approaches to modeling of propagation of ultrashort optical pulses in one and three spatial dimensions. We operate with the analytic signal formulation for the electric field rather than using the slowly varying envelope approximation, because the latter becomes questionable for few-cycle pulses. Suitable propagation models are naturally derived in terms of unidirectional approximation.     

Singularities of the Hele-Shaw flow and shock waves in dispersive media

We show that singularities developed in the Hele-Shaw problem have a structure identical to shock waves in dissipativeless dispersive media. We propose an experimental setup where the cell is permeable to a nonviscous fluid and study continuation of the flow through singularities. We show that a singular flow in this nontraditional cell is described by the Whitham equations identical to Gurevich-Pitaevski solution for a regularization of shock waves in Korteveg-de Vriez equation. This solution describes regularization of singularities through creation of disconnected bubbles.     

Coupled scalar field equations for nonlinear wave modulations in dispersive media

A review of the generic features as well as the exact analytical solutions of a class of coupled scalar field equations governing nonlinear wave modulations in dispersive media like plasmas is presented. The equations are derivable from a Hamiltonian function which, in most cases, has the unusual property that the associated kinetic energy is not positive definite. To start with, a simplified derivation of the nonlinear Schrödinger equation for the coupling of an amplitude modulated high-frequency wave to a suitable low-frequency wave is discussed. Coupled sets of time-evolution equations like the Zakharov system, the Schrödinger-Boussinesq system and the Schrödinger-Korteweg-de Vries system are then introduced. For stationary propagation of the coupled waves, the latter two systems yield a generic system of a pair of coupled, ordinary differential equations with many free parameters. Different classes of exact analytical solutions of the generic system of equations are then reviewed. A comparison between the various sets of governing equations as well as between their exact analytical solutions is presented. Parameter regimes for the existence of different types of localized solutions are also discussed. The generic system of equations has a Hamiltonian structure, and is closely related to the well-known Hénon-Heiles system which has been extensively studied in the field of nonlinear dynamics. In fact, the associated generic Hamiltonian is identically the same as the generalized Hénon-Heiles Hamiltonian for the case of coupled waves in a magnetized plasma with negative group dispersion. When the group dispersion is positive, there exists a novel Hamiltonian which is structurally same as the generalized Hénon-Heiles Hamiltonian but with indefinite kinetic energy. The above correspondence between the two systems has been exploited to obtain the parameter regimes for the complete integrability of the coupled waves. There exists a direct one-to-one correspondence between the known integrable cases of the generic Hamiltonian and the stationary Hamiltonian flows associated with the only integrable nonlinear evolution equations (of polynomial and autonomous type) with a scale-weight of seven. The relevance of the generic system to other equations like the self-dual Yang-Mills equations, the complex Korteweg-de Vries equation and the complexified classical dynamical equations has also been discussed.